Icosahedron (EntityTopic, 12)

From Hi.gher. Space

The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron.


The coordinates of an icosahedron with edge length 2 are:

(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)


  • The hypervolumes of an icosahedron with side length l are given by:
total edge length = 30l
surface area = 5√3 · l2
volume = 5(3+√5)12 · l3

Relation to other polyhedra

Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 (bilbiro) and J92 (thawro) as seen in the following animations:

The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.

Incidence matrix

Dual: dodecahedron

0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = triangle ;
3 C1a 123020 = icosahedron ;

Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind